Hybrid programs combine classical program constructs with differential equations, and thus naturally appear in a wide range of application domains, from biology and control theory to software engineering. This ability to entangle discrete and continuous behaviour, however, yields aspects unusual to computer science and renders the formal design of hybrid programs a difficult task, not properly handled by the current programming theory and practices. As a stepping stone for closing this gap, here we develop the theoretical foundations for an interpreter of hybrid programs and present a corresponding implementation – Lince. These results serve not only as basis for the implementation of typical tools of programming (e.g. debuggers and refactoring systems) but also tools specific to the hybrid domain, such as the detection of chaotic or Zeno behaviour. We also summarise Lince’s most distinctive features and illustrate its relevance for detecting design errors in hybrid programs at early development phases.